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 * (the "License"); you may not use this file except in compliance with
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package org.apache.commons.math3.distribution;

import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.special.Erf;
import org.apache.commons.math3.util.FastMath;

Implementation of the log-normal (gaussian) distribution.

Parameters: X is log-normally distributed if its natural logarithm log(X) is normally distributed. The probability distribution function of X is given by (for x > 0) 
 exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x) 

m is the scale parameter: this is the mean of the
normally distributed natural logarithm of this distribution,
s is the shape parameter: this is the standard
deviation of the normally distributed natural logarithm of this
distribution.

See Also:

 * Log-normal distribution (Wikipedia)

 * Log Normal distribution (MathWorld)
Since:
3.0/**
 * Implementation of the log-normal (gaussian) distribution.
 *
 * <p>
 * <strong>Parameters:</strong>
 * {@code X} is log-normally distributed if its natural logarithm {@code log(X)}
 * is normally distributed. The probability distribution function of {@code X}
 * is given by (for {@code x > 0})
 * </p>
 * <p>
 * {@code exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
 * </p>
 * <ul>
 * <li>{@code m} is the <em>scale</em> parameter: this is the mean of the
 * normally distributed natural logarithm of this distribution,</li>
 * <li>{@code s} is the <em>shape</em> parameter: this is the standard
 * deviation of the normally distributed natural logarithm of this
 * distribution.
 * </ul>
 *
 * @see <a href="http://en.wikipedia.org/wiki/Log-normal_distribution">
 * Log-normal distribution (Wikipedia)</a>
 * @see <a href="http://mathworld.wolfram.com/LogNormalDistribution.html">
 * Log Normal distribution (MathWorld)</a>
 *
 * @since 3.0
 */
public class LogNormalDistribution extends AbstractRealDistribution {
    
Default inverse cumulative probability accuracy. /** Default inverse cumulative probability accuracy. */
    public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;

    
Serializable version identifier. /** Serializable version identifier. */
    private static final long serialVersionUID = 20120112;

    
√(2 π) /** &radic;(2 &pi;) */
    private static final double SQRT2PI = FastMath.sqrt(2 * FastMath.PI);

    
√(2) /** &radic;(2) */
    private static final double SQRT2 = FastMath.sqrt(2.0);

    
The scale parameter of this distribution. /** The scale parameter of this distribution. */
    private final double scale;

    
The shape parameter of this distribution. /** The shape parameter of this distribution. */
    private final double shape;
    
The value of log(shape) + 0.5 * log(2*PI) stored for faster computation. /** The value of {@code log(shape) + 0.5 * log(2*PI)} stored for faster computation. */
    private final double logShapePlusHalfLog2Pi;

    
Inverse cumulative probability accuracy. /** Inverse cumulative probability accuracy. */
    private final double solverAbsoluteAccuracy;

    
Create a log-normal distribution, where the mean and standard deviation of the normally distributed natural logarithm of the log-normal distribution are equal to zero and one respectively. In other words, the scale of the returned distribution is 0, while its shape is 1. 

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. /**
     * Create a log-normal distribution, where the mean and standard deviation
     * of the {@link NormalDistribution normally distributed} natural
     * logarithm of the log-normal distribution are equal to zero and one
     * respectively. In other words, the scale of the returned distribution is
     * {@code 0}, while its shape is {@code 1}.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     */
    public LogNormalDistribution() {
        this(0, 1);
    }

    
Create a log-normal distribution using the specified scale and shape.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. 
Params:
scale – the scale parameter of this distribution
shape – the shape parameter of this distribution
Throws:
NotStrictlyPositiveException – if shape <= 0./**
     * Create a log-normal distribution using the specified scale and shape.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     *
     * @param scale the scale parameter of this distribution
     * @param shape the shape parameter of this distribution
     * @throws NotStrictlyPositiveException if {@code shape <= 0}.
     */
    public LogNormalDistribution(double scale, double shape)
        throws NotStrictlyPositiveException {
        this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Create a log-normal distribution using the specified scale, shape and
inverse cumulative distribution accuracy.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. 
Params:
scale – the scale parameter of this distribution
shape – the shape parameter of this distribution
inverseCumAccuracy – Inverse cumulative probability accuracy.
Throws:
NotStrictlyPositiveException – if shape <= 0./**
     * Create a log-normal distribution using the specified scale, shape and
     * inverse cumulative distribution accuracy.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     *
     * @param scale the scale parameter of this distribution
     * @param shape the shape parameter of this distribution
     * @param inverseCumAccuracy Inverse cumulative probability accuracy.
     * @throws NotStrictlyPositiveException if {@code shape <= 0}.
     */
    public LogNormalDistribution(double scale, double shape, double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        this(new Well19937c(), scale, shape, inverseCumAccuracy);
    }

    
Creates a log-normal distribution.
Params:
rng – Random number generator.
scale – Scale parameter of this distribution.
shape – Shape parameter of this distribution.
Throws:
NotStrictlyPositiveException – if shape <= 0.
Since:
3.3/**
     * Creates a log-normal distribution.
     *
     * @param rng Random number generator.
     * @param scale Scale parameter of this distribution.
     * @param shape Shape parameter of this distribution.
     * @throws NotStrictlyPositiveException if {@code shape <= 0}.
     * @since 3.3
     */
    public LogNormalDistribution(RandomGenerator rng, double scale, double shape)
        throws NotStrictlyPositiveException {
        this(rng, scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Creates a log-normal distribution.
Params:
rng – Random number generator.
scale – Scale parameter of this distribution.
shape – Shape parameter of this distribution.
inverseCumAccuracy – Inverse cumulative probability accuracy.
Throws:
NotStrictlyPositiveException – if shape <= 0.
Since:
3.1/**
     * Creates a log-normal distribution.
     *
     * @param rng Random number generator.
     * @param scale Scale parameter of this distribution.
     * @param shape Shape parameter of this distribution.
     * @param inverseCumAccuracy Inverse cumulative probability accuracy.
     * @throws NotStrictlyPositiveException if {@code shape <= 0}.
     * @since 3.1
     */
    public LogNormalDistribution(RandomGenerator rng,
                                 double scale,
                                 double shape,
                                 double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        super(rng);

        if (shape <= 0) {
            throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
        }

        this.scale = scale;
        this.shape = shape;
        this.logShapePlusHalfLog2Pi = FastMath.log(shape) + 0.5 * FastMath.log(2 * FastMath.PI);
        this.solverAbsoluteAccuracy = inverseCumAccuracy;
    }

    
Returns the scale parameter of this distribution.
Returns:
the scale parameter/**
     * Returns the scale parameter of this distribution.
     *
     * @return the scale parameter
     */
    public double getScale() {
        return scale;
    }

    
Returns the shape parameter of this distribution.
Returns:
the shape parameter/**
     * Returns the shape parameter of this distribution.
     *
     * @return the shape parameter
     */
    public double getShape() {
        return shape;
    }

    
{@inheritDoc} For scale m, and shape s of this distribution, the PDF is given by 

0 if x <= 0,
exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x) otherwise.

/**
     * {@inheritDoc}
     *
     * For scale {@code m}, and shape {@code s} of this distribution, the PDF
     * is given by
     * <ul>
     * <li>{@code 0} if {@code x <= 0},</li>
     * <li>{@code exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
     * otherwise.</li>
     * </ul>
     */
    public double density(double x) {
        if (x <= 0) {
            return 0;
        }
        final double x0 = FastMath.log(x) - scale;
        final double x1 = x0 / shape;
        return FastMath.exp(-0.5 * x1 * x1) / (shape * SQRT2PI * x);
    }

    
{@inheritDoc} See documentation of density(double) for computation details. /** {@inheritDoc}
     *
     * See documentation of {@link #density(double)} for computation details.
     */
    @Override
    public double logDensity(double x) {
        if (x <= 0) {
            return Double.NEGATIVE_INFINITY;
        }
        final double logX = FastMath.log(x);
        final double x0 = logX - scale;
        final double x1 = x0 / shape;
        return -0.5 * x1 * x1 - (logShapePlusHalfLog2Pi + logX);
    }

    
{@inheritDoc} For scale m, and shape s of this distribution, the CDF is given by 

0 if x <= 0,
0 if ln(x) - m < 0 and m - ln(x) > 40 * s, as in these cases the actual value is within Double.MIN_VALUE of 0, 
1 if ln(x) - m >= 0 and ln(x) - m > 40 * s, as in these cases the actual value is within Double.MIN_VALUE of 1,
0.5 + 0.5 * erf((ln(x) - m) / (s * sqrt(2)) otherwise.

/**
     * {@inheritDoc}
     *
     * For scale {@code m}, and shape {@code s} of this distribution, the CDF
     * is given by
     * <ul>
     * <li>{@code 0} if {@code x <= 0},</li>
     * <li>{@code 0} if {@code ln(x) - m < 0} and {@code m - ln(x) > 40 * s}, as
     * in these cases the actual value is within {@code Double.MIN_VALUE} of 0,
     * <li>{@code 1} if {@code ln(x) - m >= 0} and {@code ln(x) - m > 40 * s},
     * as in these cases the actual value is within {@code Double.MIN_VALUE} of
     * 1,</li>
     * <li>{@code 0.5 + 0.5 * erf((ln(x) - m) / (s * sqrt(2))} otherwise.</li>
     * </ul>
     */
    public double cumulativeProbability(double x)  {
        if (x <= 0) {
            return 0;
        }
        final double dev = FastMath.log(x) - scale;
        if (FastMath.abs(dev) > 40 * shape) {
            return dev < 0 ? 0.0d : 1.0d;
        }
        return 0.5 + 0.5 * Erf.erf(dev / (shape * SQRT2));
    }

    
{@inheritDoc}
Deprecated:
See RealDistribution.cumulativeProbability(double, double)/**
     * {@inheritDoc}
     *
     * @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
     */
    @Override@Deprecated
    public double cumulativeProbability(double x0, double x1)
        throws NumberIsTooLargeException {
        return probability(x0, x1);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double probability(double x0,
                              double x1)
        throws NumberIsTooLargeException {
        if (x0 > x1) {
            throw new NumberIsTooLargeException(LocalizedFormats.LOWER_ENDPOINT_ABOVE_UPPER_ENDPOINT,
                                                x0, x1, true);
        }
        if (x0 <= 0 || x1 <= 0) {
            return super.probability(x0, x1);
        }
        final double denom = shape * SQRT2;
        final double v0 = (FastMath.log(x0) - scale) / denom;
        final double v1 = (FastMath.log(x1) - scale) / denom;
        return 0.5 * Erf.erf(v0, v1);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected double getSolverAbsoluteAccuracy() {
        return solverAbsoluteAccuracy;
    }

    
{@inheritDoc} For scale m and shape s, the mean is exp(m + s^2 / 2). /**
     * {@inheritDoc}
     *
     * For scale {@code m} and shape {@code s}, the mean is
     * {@code exp(m + s^2 / 2)}.
     */
    public double getNumericalMean() {
        double s = shape;
        return FastMath.exp(scale + (s * s / 2));
    }

    
{@inheritDoc} For scale m and shape s, the variance is (exp(s^2) - 1) * exp(2 * m + s^2). /**
     * {@inheritDoc}
     *
     * For scale {@code m} and shape {@code s}, the variance is
     * {@code (exp(s^2) - 1) * exp(2 * m + s^2)}.
     */
    public double getNumericalVariance() {
        final double s = shape;
        final double ss = s * s;
        return (FastMath.expm1(ss)) * FastMath.exp(2 * scale + ss);
    }

    
{@inheritDoc}
The lower bound of the support is always 0 no matter the parameters.
Returns:
lower bound of the support (always 0)/**
     * {@inheritDoc}
     *
     * The lower bound of the support is always 0 no matter the parameters.
     *
     * @return lower bound of the support (always 0)
     */
    public double getSupportLowerBound() {
        return 0;
    }

    
{@inheritDoc}
The upper bound of the support is always positive infinity
no matter the parameters.
Returns:
upper bound of the support (always Double.POSITIVE_INFINITY)/**
     * {@inheritDoc}
     *
     * The upper bound of the support is always positive infinity
     * no matter the parameters.
     *
     * @return upper bound of the support (always
     * {@code Double.POSITIVE_INFINITY})
     */
    public double getSupportUpperBound() {
        return Double.POSITIVE_INFINITY;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportLowerBoundInclusive() {
        return true;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportUpperBoundInclusive() {
        return false;
    }

    
{@inheritDoc}
The support of this distribution is connected.
Returns:
true/**
     * {@inheritDoc}
     *
     * The support of this distribution is connected.
     *
     * @return {@code true}
     */
    public boolean isSupportConnected() {
        return true;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double sample()  {
        final double n = random.nextGaussian();
        return FastMath.exp(scale + shape * n);
    }
}